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Factoring Algebra

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Math factoring is very much useful in almost all the fields of mathematics. In factoring Algebra, factors are any algebraic expression which divides another algebraic expression where the remainder is zero. Similarly in case of Numbers we use numbers instead of algebraic expressions.For instance: 4 and 2 are the factors of 8 which means that both the numbers divides 8 completely as 8 ÷ 2 = 4 and 8 ÷ 4 = 2.

We can also define factors as the numbers which we multiply so that we can get other number. For example factors of 12 are 3 and 4, because 3*4 = 12. Some numbers may have many factors. Example is 16 can be factored as 1*16, 2*8, or 4*4.
The number which can only be factorized as 1 and itself then that number is called prime number. For example 2, 3, 5, 7, 11, 13 . . . . are the Prime Numbers. The number 1 is always ignored in case of Factorization because it comes always in almost everything and it do not considered as a prime number.
The factorization of prime numbers does not include 1. It includes all the copies of each and every prime factor. For example: Prime factorization of 8 = 2*2*2 that is not only 2. Here 2 is the only factor of 8 but it is not sufficient means we need three copies of 2 so if we multiply 2 by itself 3 times we get back 8; therefore the prime factorization of 8 includes 3 copies of 2.
Again if we look at the prime factorization it will only consider the prime factors not the products of those factors. For example: 2*2 is 4 and 4 is a divisor of 8 but 4 is not considered as a prime factor of 8. Because 8 is not equal to 2*2*2*4.
Suppose in factoring Math the task is to get the prime factorization of 24. Now if we only collect all the divisors of 24 as 1, 2, 3, 4, 6, 8, 12, and 24 and we multiply all the divisors of 24 as 1*2*3*4*6*8*12*24 will result in 331776 which are not 24 so it is not the right way. So we use prime factorization, either the problem needs it or not to avoid such over duplication or multiplication. So for this we can find the prime factor of 24 by dividing 24 by the smallest prime number which is 2 i.e. 24 / 2 = 12. Now again divide the result 12 by the smallest prime number 2 as 12 / 2 = 6; again divide 6 by 2 as 6 / 2 = 3 and here we get a prime number that is 3 so we do not need to further divide this because the prime numbers has factors 1 and itself; so we are done. So the factors of 24 are 2, 2, 2 and 3.
Find the prime factor of 1008
Solution:
1002 ÷ 2 = 504
504 ÷ 2 = 252
252 ÷ 2 = 126
126 ÷ 2 = 61
So it has solution 1002 = (504) (252) (126) (61).

GCF and Prime Factorization

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To find the factors of any number means to write the number in form of its multiples. If the factors of the number are all prime Numbers, then we say that the factors are prime factors. For this we go on breaking the number in form of its multiples until we get all its factors as the Prime Numbers. E.g.: Prime factors of 20 = 2 * 5 * 2 * 1. Here if we write the factors as 5 * 4, then 4 is not prime, or if we write 10 * 2, then 10 is not a prime number. So they can be called factors but not prime.

Greatest common factors of any two numbers are the factors which are common for any given two of more numbers. We also call it Highest common factor or HCF. We can find GCF using prime Factorization method and also by repeated division method. We get the same result in both the cases. But here in this unit we are using prime factorization method, in which we first find the prime factors of all the numbers whose greatest common factor is to be calculated and then place the common factors of all the numbers together and then find their product.

Solving Quadratic Equations

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Any equation which can be written in the form: ax2 + bx + c = 0 where we have a, b, and c as the real Numbers and value of ‘a’ is not equal to zero, than it is called as Quadratic Equation. Here we will learn how to solve quadratic equation. Let us consider a real number ‘α’ (alpha) which is the root of the quadratic equation ax2 + bx + c = 0, a≠0, if aα2+ bα+ c = 0. From the above statement we conclude that:
If ‘α’ is the root of the quadratic equation ax2 + bx + c = 0, then it means that x = α is the solution of the quadratic equation ax2 + bx + c = 0.
By the term solving quadratic equations, we Mean to find the roots of the given quadratic equation. We also call these roots of the quadratic equation as the zero of the polynomial, which means that if this value of ‘x’ is placed in the given quadratic equation, and we solve quadratic equations the result is zero.
There are different methods of solving the quadratic equations. First method of solving the quadratic equation and to find its roots is by Factorization. Let us take an example: 6x2 - x – 2 = 0,
Here we observe that on comparing this equation by the standard quadratic equation we get : a = 6, b = -1 and c = -2.
Now the middle term of the above expression is resolved in such a way that the sum is –x and the product is (-2 * 6x2) = -12x2,
So we write it as 6x2 - 4x + 3x – 2 = 0
Or 2x (3x – 2) + 1 (3x-2) = 0,
Or (3x-2) (2x +1) = 0,
So we get either (3x-2) = 0 or (2x +1) = 0,
So we have 3x = 2 or 2x = 1,
So we get x = 2/3 or x = ½,
Hence we have x = 2/3 or x = ½ as the roots of the quadratic equation 6x2 - 4x + 3x – 2 = 0,
There is another method to solve the quadratic equations by the help of finding the value of discriminate (D), where D = root (b2 – 4 a. c) and once ‘D’ is known we get the two roots as
α= -b + √ (D)/2a and Β= -b + √ (D)/2a.

Factor Trinomials

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It is just reverse process of multiplying. Factoring Trinomials is always in the form of x2 + bx + c. It is just opposite of Foil Method i.e. the process which is used to multiply two binomial values. Distributive property explains how to multiply a single value such as 8 by a binomial such as (6 + 8x) which is represented as 8 (6 + 8x).
If there is an expression which is consist of one binomials value in parentheses. Using the foil method we can simply this equation which is given as a Trinomials result.
5p2 - 18s + 19s – 85,
5p2 + k – 85,
This expression explains how to factor Trinomials.
The binomial expression which is coming after factoring as called factor trinomials. In last trinomial expression which is x2 + 8x + 12 has two factors “x+6” and “x+2”, these are factors of trinomials for this expression.
Now we will see factoring a trinomial, it can be understood using following steps.
We know factoring is a reverse process of multiplying. When we multiply two factors, answer is the original function.
Let us take an example to know about the factoring of trinomial.
x2 + 9x + 14 is a trinomial, we find the factors of this trinomial.
It will be factored into two binomials:
(x +?) (x +?)
Because first term of binomial is factor of x2, it comes from multiplying ‘x’ with ‘x’ and second binomial term comes from factoring of 14.
Now the factor of 14 is 2 and 7, and the addition of this factor is 9, which is second term. So they can be written as
(x 2) (x 7),
Because there is a + sign between terms of trinomial so they will write as
(x +2) (x+7),
These are the factors of trinomial x2 + 9x + 14.
If there is – sign between first and second term like x2 - 9x + 14, then factors for this trinomial are
(x -2) (x -7),
Factoring trinomials is a reverse process of multiplying. It means if we multiply 5 with 9, the answer is 45, it means 5 and 9 are two factors of 45.
In trinomial factoring two factors are the outcomes, because trinomial has maximum two degree of power.

Factoring GCF

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Word gcf means to find the greatest common factor. Factoring gcf, also means to find the highest common factor. When we find the factors of any number it means that we need to find the Numbers which are divisible by the given number. If we write that 3 is the factor of 12, it means that 12 is completely divisible by 3 and does not leave any remainder. Factoring by gcf, can be done by finding the prime factors of the given numbers and then finding the product of the common factors among the given factors. First let us discuss how to find the prime factors of the given number.
In order to find the prime factors of the given number, we will write the number as the product of its factors which are prime. Now let us make it more clear with the help of the example. If we need to find the factors of 24, we can write 24 = 12 * 2, but we observe that 12 is not the prime number. Thus they are the factors of 24, but not the prime factors. We write prime factors of 24 as 24 = 2 * 2 *2* 3 * 1. Now we find that all these factors are prime. Similarly if we find the prime factors of 27, it will be, 27 = 3 * 3 * 3 * 1. Again we find that all these factors are Prime Numbers. Now if we need to find the gcf, we observe that the common factors of 24 and 27 are 3 and 1. Thus we write that the gcf of 24 and 27 are 3 * 1 = 3.
The full form of gcf is greatest common factor. It simply means to find the largest number which divides both the numbers. There can be other numbers, which may divide the numbers individually, but we are here talking about the greatest number which will be dividing both the given numbers. In the same way we can find the greatest common divisor for more than two numbers. In such case, we will again proceed in the same way. We will first find the prime factors individually for all the numbers separately, then Circle the numbers which are common factors for all the numbers and then multiplying them will give us the gcf of the given Set of numbers. This is the process of factoring out the gcf.

Factoring Difference of Square

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When an algebraic expression can be written in the form of the product of two or more expressions, then each of these expressions is called as the factor of the given expression. We find the factors of the algebraic expression with the aim to simplify the expression and it will help us to make the calculations easier and fast when the variables in the expression are replaced by the numerical constants. We need to learn the method of solving and factorizing the algebraic expressions using different techniques. These techniques vary from time to time and the choice of selecting the method of factorizing needs lots of observations and practice.
Solving algebraic expressions or mathematical equations can be made easy by simplifying it and then converting them in the form of the standard products. Some of the equations are expressed as the relations named squares of the sum as (a + b)2, squares of the difference as (a - b)2 and difference of the squares which is represented in form of a2 – b2.
In this unit we will learn about factoring difference of squares. Suppose we have any expression (a2 – b2) and we have to factor difference of squares, for this we need to find the factors of the difference of the squares of ‘a’ and ‘b’. Thus for factoring differences of squares we will write,
(a2 – b2) as ( a + b) * ( a – b),
SO the formula for (a2 – b2) = (a + b) * (a – b),
This formula helps us to solve various mathematical expressions too, in order to simplify the expressions, we will first relate the expression with the above standard difference of the squares and it will become easy for us to calculate the values which otherwise looks difficult and complicated.
Let us take the example for solving 102 * 98. It looks time consuming and difficult activity to multiply the above given Numbers. But if we write the above expression as the sum ad the product relation it becomes:
102 * 98 = (100 + 2) * (100 – 2),
It is now visible in the form of the equation (a2 – b2) = ( a + b) * ( a – b),
So relating the solution with the above equation, we write it as: 1002 - 22,
Or = 10000 – 4 = 99996.

Factoring Perfect Square Trinomials

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Perfect Square trinomial is a trinomial which when factored gives two identical factors or we can say that the factors are same.
For example we have an equation p2 + 8p + 16; then we can write it as. (p + 4) (p + 4) or we can write it as (p + 4)2.
For factoring a perfect square trinomial we need to follow some steps:
Step1: First we take an equation for factoring perfect square trinomials.
Step2: Multiply the square and constant term and break middle term such that when broken terms are multiplied gives back the product of squared term and constant term.
Step3: Now group the factors by taking common terms out and we get the factors.
Now we will see factoring perfect square trinomial with the help of an example shown below:
We know that factoring is a reverse process of multiplying the expression. Now we see the process of factoring a trinomial.
⇒ P2 + 10P + 25 is a trinomial, we will find the factor of this expression, after factoring we get two binomials:
(P ?) (P ?)
As the first term of binomial is factor of P2 so it comes when we multiply ‘P’ with ‘P’ and second binomial term comes from factoring of 25.
The factors of 25 are 5 and 5, and the addition of these factors is 10, which is middle term. So they can be written as:
⇒ (P 5) (P 5)
There is a ‘+’ sign between terms of trinomial so they will be written as:
⇒ (P + 5) (P + 5)
These are the factors of trinomial P2 + 10P + 25.
If negative sign is present between first and second then we can write the expression as
P2 – 10P + 25, then factors for this trinomial are:
⇒ (P - 5) (P - 5)
This is all about factoring a perfect square trinomial.